Computers Could Resolve Mathematics’ Biggest Controversy

A major clash in the world of mathematics may see resolution thanks to computers, potentially bringing an end to a decade-long dispute surrounding a complex proof.

It all began in 2012 when Shinichi Mochizuki, a mathematician from Kyoto University in Japan, shocked the mathematical community with his extensive 500-page proof of the ABC conjecture. This conjecture stands as a significant unsolved issue at the very essence of number theory. Mochizuki’s proof relied on an intricate and obscure framework that he developed, known as Interuniversal Teichmuller (IUT) theory, which proved challenging for even seasoned mathematicians to grasp.

The ABC conjecture, which has been around for over 40 years, presents a seemingly straightforward equation involving three integers: a + b = c, investigating the relationships among the prime numbers that constitute these values. The conjecture offers profound insights into the fundamental interactions of addition and multiplication, with ramifications for other renowned mathematical conjectures, including Fermat’s Last Theorem.

Given these potential consequences, mathematicians initially expressed excitement over verifying the proof. However, Mochizuki noted that early attempts faced challenges and more focus was needed on understanding his findings. In 2018, two distinguished German mathematicians, Peter Scholze from the University of Bonn and Jakob Stix from Goethe University in Frankfurt, announced that they had found possible flaws in the proof.

Mochizuki, however, dismissed these critiques. Lacking a central authority to arbitrate the debate, the credibility of the IUT theory has split the mathematical community into opposing factions, with one side comprising a small collective of researchers aligned with Mochizuki and the Kyoto Institute for Mathematical Sciences, where he teaches.

Now, Mochizuki has suggested a path forward to resolve the deadlock. He proposes transferring proofs from their existing mathematical notation, intended for human comprehension, to a programming language known as Lean, which can be validated and checked by computers.

This approach, known as formalization, represents a promising area of research that could revolutionize the practice of mathematics. Although there have been earlier suggestions for Mochizuki to formalize his proof, this marks the first time he has publicly indicated plans to advance this initiative.

Mochizuki was unavailable for comment on this article. However, in recent reports, he asserted that Lean would be an excellent tool for clarifying certain disputes among mathematicians that have hindered acceptance of his proof. He stated, “This represents the best, and perhaps only, way to achieve significant progress in liberating mathematical truth from social and political constraints.”

Mochizuki became convinced of the advantages of formalization after attending a conference on Lean in Tokyo last July, particularly impressed by its capacity to manage the mathematical structures essential to his IUT theory.

This could be a vital step in overcoming the current stalemate, noted Kevin Buzzard from Imperial College London. “If it’s articulated using Lean, that’s not strange at all. Much of what’s found in papers is written in unusual terms, so being able to express it in Lean means that this unusual language has become universally defined,” he explains.

“We seek to understand why [of IUT], and we’ve been awaiting clarity for over a decade,” remarked Johann Kommelin from Utrecht University in the Netherlands. “Lean will aid in uncovering those answers.”

However, both Buzzard and Kommelin acknowledge that formalizing IUT theory is an immense challenge, necessitating the conversion of a series of mathematical equations that currently exist only in a human-readable format. This effort is anticipated to be the largest formalization endeavor ever attempted, often requiring teams of specialists and taking months or even years.

This daunting reality may dissuade the limited number of mathematicians capable of undertaking this project. “Individuals will need to decide whether they are willing to invest significant time in a project that may ultimately lead to failure,” Buzzard remarked.

Even if the mathematicians succeed in completing the project and the Lean code indicates that Mochizuki’s theorem is consistent, disputes about its interpretation could still arise among mathematicians, including Mochizuki himself, according to Kommelin.

“Lean has the potential to make a significant impact and resolve the controversy, but this hinges on Mochizuki’s genuine commitment to formalizing his work,” he adds. “If he abandons it after four months, claiming ‘I’ve tried this, but Lean is too limited to grasp my proof,’ it would just add another chapter to the long saga of social issues persisting.”

Despite Mochizuki’s enthusiasm about Lean, he concedes with his critics that interpreting the meaning of the code might lead to ongoing disputes, expressing that Lean “does not appear to be a ‘magic cure’ for completely resolving social and political issues at this stage.”

Nevertheless, Buzzard remains optimistic that the formalization project, especially if successful, could propel the decade-old saga forward. “You can’t contest software,” he concludes.